Uncertainty and sensitivity analysis for hydraulic models with dependent inputs

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Uncertainty and sensitivity analysis for hydraulic models with dependent inputs

Lucie Pheulpin, Vito Bacchi, Nathalie Bertrand

To cite this version:

Lucie Pheulpin, Vito Bacchi, Nathalie Bertrand. Uncertainty and sensitivity analysis for hydraulic models with dependent inputs. European Geosciences Union General Assembly, EGU, Apr 2019, VIENNE, Austria. 2019. �hal-02635597�

IRSN

DE RADIOPROTECTION ET DE SÛRETÉ NUCLÉAIRE

Uncertainty and sensitivity analysis for hydraulic models with dépendent inputs

Lucie Pheulpin1, Vito Bacchi1 and Nathalie Bertrand1

Institut de Radioprotection et de Sûreté Nucléaire, Fontenay-aux-Roses, France Contact : [email protected]

1. Introduction

Nowadays, flooding hazard is usually assessed through numerical modelling, generally affected by uncertainties. Uncertainty Quantification (UQ) and Global Sensitivity Analysis can be useful tools to improve the quantification of the flooding hazard.

Traditionally, to perform these kinds of analyses, the input parameters are supposed to be independent, which is not always the case. In the framework of the NARSIS European Research-project, our objective is to develop a methodology to perform UQ and GSA by considering dependent inputs. This methodology will be applied to the Loire River 2D hydraulic model, currently under construction.

However, before applying the general methodology presented here, we tested it on a very simplified model of river flood inundation.

2. Methodology

Step A: Problem specification

Input parameters:

> Fixed: Time step, grid resolution, etc.

>

• Hydraulic parameters:

hydrograph parameters, Strickler coefficient, etc.

• Breach parameters: length, depth, time formation, etc.

Independent parameters or not?

Variables of interest

> Water levels at certain location in the flood plain (e.g. near the

breaches)

Quantities of interest

> Probability, variance, etc.

Hydraulic and levee breach modelling: Example for the Loire River

> 50 km-long reach modelling, between Gien and Orléans

> 2D modelling with Telemac-2D

> Numerous levees along this reach with known historical breaches

Example of Loire River modelling

Dam pi erre

A Nuclear Power Plant

^ Hydrometric station

T Historical breaches

Modelling area

Step B: Uncertainty sources quantification

For all parameters, definition of:

> Parameter bounds

> Parameter distribution laws For dependent parameters:

> Groups of parameters identification

> Copula selection (e.g.

normal copula) adapted to each group of parameter and definition of the

correlation coefficients (r)

> Construction of multivariate distributions

Exampb of a normal copula cumulative distribution fUncton

Step C: Uncertainty Quantification (UQ)

Random sample of input parameters with the computational environment Prométhée (e.g. with a Monte-Carlo method)

> For independent parameters: inside their distributions laws

> For dependent parameters: inside the multivariate distributions coming from copulas

^ Construction of histograms, boxplots, etc. of outputs

Step D: Global Sensitivity Analysis (GSA) #

Prométhée - TELEMAC-2D

<---

open TELEMAC-MASCARET

The mathematically superior suite of solvers

Variance based method: computation of Sobol indices (1st and total order)

> With a FAST (Fourier Analysis Sensitivity Test) method for independent parameters

> Calculation of multidimensional sensitivity indices for dependent parameters (Jacques, Lavergne, et al. 2006)

Screening method: computation of sensitivity indices (elementary mean and standard deviation) with Morris method

> With a classic Morris method for independent parameters

> With an extension of the Morris method which integrates dependency through copulas for dependent parameters (Jene et al.,2018)

^

3. Test case: simplified model of river flood inundation

Model description Model equations

> Based on simplified 1D hydro-dynamical equations of Saint- Venant, considering uniform and constant flowrate and large

rectangular sections (used in Iooss and Lemaître, 2015)

> Simulation of river water level (h) and comparison with levee

height (Hd) > Random sampling of 10,000 parameter combinations in

> 8 input parameters: 3 groups of 2 inputs (Q/Ks, Zv/Zm, L/B) the univariate (independent inputs) or multivariate and 2 independent ones: (Cb and Hd) (dependent inputs) distributions

h = ^Q

0.6

with S = Zv + h — Hj — C BK^s

d 'b

Uncertainty sources quantification

Inputs Symbols Units PDF

Maximal annual flow rate Q m3/s Truncated Gumbel Strickler coefficient Ks - Truncated Normal River downstream level Zv m Triangle

River upstream level Zm m Triangle

Levee height Hd m Uniform

Bank level Cb m Triangle

Length of the river stretch L m Triangle

River width B m Triangle

UQ for independent and dependent parameters

Dependent inputs ^ 3 normal copulas: Q/Ks (r = 0.5) ; Zv/Zm (r = 0.3) ; L/B (r = 0.3)

Normal copula Q/K_s Outputs Distribution Independent inputs *

Dependent inputs *

GSA

CO COCD °

O

TD C

Independent §5 parameters 1

Od

Dependent parameters

C/) CD O

d^g_>.

’>

(f)C CD

CO co

d

od

Variance-based methods (Sobol indices)

FAST method

II

Q

II

Ks

I î

Zv Zm Hd

Input variables

Cb

• first order indices

• total indices

~r~L ^~~r~B

io

cO

.55 o

> ■ (D

-o

CO U

c.CO

(/)

LO

Calculation of multidimensional senstvity indices

h

" î il

Q/Ks Zv/Zm Hd

Multidimensional variables

Cb

first order indices total indices

L/B

c o o

Screening methods (Morris)

Classic Morris method _•Ks

•Q

_ •Zm

- #l?B •Cb •Hd •Zv

0.0 0'5 T0 T5

Absolute mean

2.0 2^5

•Q-Ks Mode Morris mehod oonsdtemg

dependencies through copulas

•L-B •Cb Hd®Zv-Zm

T---1--- i---1--- 1---T

0.0 0.5 1.0 1.5 2.0 2.5

Absolute mean

> In this example, the choice of the copula has very few impact on the outputs and there is almost no difference between the distribution of outputs by considering certain inputs dependent or not.

> The GSA methods show that some parameters (e.g. Zm) can have more influence once included in a group than considered independent.

4. Conclusion and perspectives

> In the test case, the copulas and their correlation coefficients are defined arbitrarily. In the reality (i.e. in hydraulic models), it is necessary to test different types of copulas and different groups of parameters inside copulas, on observed data, and to validate them with a Cramer-von-Mises test for example.

> The UQ and GSA tools used for the test case were coded with R and now they must be included in the computational environment Prométhée.

> Once the Telemac-2D Loire model achieved, it will be coupled with Prométhée to process UQ and GSA on hydraulic parameter and on levee breach parameters. Finally, the whole point of our research is to better estimate the flooding hazard.

Références

J. Jacques, C. Lavergne, and N. Devictor, "Sensitivity analysis in presence of model uncertainty and correlated inputs", Reliability Engineering & System Safety, vol. 91, no 10-11, p. 1126-1134, oct. 2006.

M. Jene, D. E. Stuparu, D. Kurowicka, and G. Y. El Serafy, "A copula-based sensitivity analysis method and its application to a North Sea sediment transport model", Environmental Modelling & Software, vol. 104, p. 1-12, june 2018.

B. Iooss and P Lemaître, "A Review on Global Sensitivity Analysis Methods", in Uncertainty Management in Simulation-Optimization of Complex Systems, vol. 59, G. Dellino et C. Meloni, Éd. Boston, MA: Springer US, 2015, p. 101-122.

Work carried out within the European project NARSIS (New Approach to Reactor Safety ImprovementS)

Photography encouraged